Subgroups — #12514

Order 256.

123 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2241	C₂	1	4	yes	maximal
2	#1713	C₂	1	1	yes	maximal, characteristic
3	#1710	C₂	1	1	yes	maximal, characteristic
4	#1311	C₂	1	1	yes	maximal, characteristic
5	#500	—	2	2	no	—
6	#500	C₂²	1	2	yes	—
7	#494	C₂²	1	2	yes	—
8	#487	—	2	2	no	—
9	#358	—	2	8	no	—
10	C₄×C₁₆	C₂²	1	2	yes	—
11	C₈²	C₂²	1	1	yes	characteristic
12	C₂×Q₁₆	—	2	4	no	—
13	C₄⋊Q₈	—	2	2	no	—
14	C₄⋊Q₈	—	2	2	no	—
15	C₂×C₁₆	D₄	1	4	yes	—
16	#107	—	2	4	no	—
17	#107	—	4	4	no	—
18	#103	—	4	4	no	—
19	C₄×C₈	D₄	1	2	yes	—
20	C₄×C₈	C₂³	1	1	yes	characteristic
21	C₂×Q₈	—	4	2	no	—
22	C₂×Q₈	—	4	2	no	—
23	Q₁₆	—	4	8	no	—
24	C₂×C₈	D₈	1	4	yes	—
25	C₂×C₈	C₂×D₄	1	2	yes	—
26	C₄⋊C₄	—	4	2	no	—
27	C₄⋊C₄	—	4	2	no	—
28	C₄⋊C₄	—	4	4	no	—
29	C₄²	C₂×D₄	1	1	yes	characteristic
30	C₁₆	—	2	4	no	—
31	Q₈	—	8	4	no	—
32	Q₈	—	8	4	no	—
33	C₂×C₄	—	8	2	no	—
34	C₂×C₄	—	8	2	no	—
35	C₂×C₄	#127	1	1	yes	characteristic
36	C₂×C₄	C₂×D₈	1	2	yes	—
37	C₈	—	2	4	no	—
38	C₈	#135	1	4	yes	—
39	C₂²	#493	1	1	yes	characteristic
40	C₄	—	16	2	no	—
41	C₄	—	16	2	no	—
42	C₄	#495	1	2	yes	—
43	C₄	#510	1	4	yes	—
44	C₂	#1709	1	1	yes	characteristic
45	C₂	#2251	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.